The free vibrations of a system are governed by the differential equation 

\[ 2 \ddot{x}+40 \dot{x}+1800 x=0 \]

with initial conditions \(x(0)=0.001 \mathrm{~m}\) and \(\dot{x}(0)=3 \mathrm{~m} / \mathrm{s}\). Calculate or specify the following.

(a) The natural frequency, \(\omega_{n}\)
(b) The damping ratio, \(\zeta\)
(c) Whether the system is undamped, underdamped, critically damped, or overdamped
(d) The undamped period, \(\mathrm{T}\)
(e) The frequency in \(\mathrm{Hz}, f\)
(f) The damped natural frequency (if appropriate), \(\omega_{d}\)
(g) The logarithmic decrement (if appropriate), \(\delta\)
(h) The amplitude, \(A\)
(i) The phase between the response and a pure sinusoid (if appropriate), \(\phi\)
(j) The free response of the system